An example related to the Monotone Convergence Theorem

Question

Let $f_n=\frac{1}{n}\chi_{[0,n]}$, which converge a.e. to $f=0$. Then $$\int f d\lambda \neq \lim\int f_n d\lambda$$

Why does this not contradict the Monotone Convergence Theorem?

First, this sequence is decreasing, making the hypothesis of monotone convergence theorem is not fulfilled.

Answer 1

The monotone convergence theorem requires $f_1(x) \leq f_2(x) \leq \cdots$ for all $x$.

But $$f_1(x) = \begin{cases} 1 & \text{if } x \in [0, 1], \\ 0 & \text{otherwise,} \end{cases}$$

and $$f_2(x) = \begin{cases}\frac12 & \text{if } x \in [0, 2], \\ 0 & \text{otherwise}\end{cases}$$

Notice that $f_1(0) = 1 \not\leq \frac12 = f_2(0)$. So the hypothesis fails.